332 XV NORMED ALGEBRAS AND SPECTRAL THEORY (use (a)). Deduce that CT(XI, ^2) = a(x* • X<) and that Deduce from this inequality that, if %lt #2 , Xz are tnree characters of A, we have o(y *"*3'= and hence that the relation a(^', x") < 1 between characters ^', x" of A is an equiva- lence relation on X(A). The equivalence classes for this relation are called Gleason parts of X(A). (c) Let u e A be such that p(u) ^ 1, and let ^i, X* ^e two characters of A, such that Xi(w) = 0. Show that there exists a complex number A such that |A| < 1 and such that, if we put v = (A — w)(l — Aw)~l e A, then xi(v) = A and x2(t>) = —A. Deduce that }< 4|XI(,) *2 ; = (d) If xi, X2 are two characters of A, put T(xi,x2) = SUP \ ue A, /)(u)< 1 Show that (use (c)). The equivalence relation defined in (b) is therefore equivalent to r(x', x") < 2. (e) Let we A be such that ^x(u) = ^ ^or a11 Xe*(A). Show that, for any two characters Xi > X2 > we nave (Apply (*) to the element e~ru, where / > 0, and let t tend to 0.) 19. With the hypotheses of Problem 8, let xi, X* ^e two characters of B. (a) Let ftj.,/^2 be representative measures for %i,X2 respectively (Problem 9). If cr(xi, ^2) = 1 (Problem 18), show that ^ and ju,2 are disjoint (13.18). (Write ft2 ^ h - fji! + v where v is disjoint from /xj, and by majorizing |^2(w) — XiWI for u e B and ||u|| ^1, show that the hypothesis implies that I (h + 1) d^ g I \h — 1| flf/x!.) (b) If <T(XI, x^)< 1> show that there exist representative measures pi for ^i and fju2 for X2 such that (Put c== (1 — o-(^1? ^2)(1 + o-(xr, Xz))"1- Use Problem 18(e) and the Hahn-Banach(/x) and conversely (Problem 15(e)). A function